Data Characteristics

Central Tendency

• Mean
• Median
• Mode

Range:

• Max
• Min
• Quantiles
• Outlier

Dispersion

• Variance
• Standard Deviation

Skew

Mean

Population Mean - $\bar{x}=\frac{1}{n}\sum _{i=1}^n{x}_i$ Sample Mean - $\mu =\frac{\sum x}{N}$ Weighted Mean - $\bar{x}=\frac{{\sum }_{i=1}^n{w}_i{x}_i}{{\sum }_{i=1}^n{w}_i}$ Trimmed Mean - mean after removing outliers

Median

Suitable for skewed data Expensive to compute for large dataset, solution = approximate for grouped data

Mode

Most occurring value in dataset Unimodal - moderately skewed data: $mean-mode=3\cdot (mean-median)$ Under Normal Distribution : $mean=mode=median$

Multi Modal

Bimodal

Two Peaks Result of combining 2 different processes, eg. body mass of males and females

Trimodal

Dispersion

Variance ( $\sigma$ | s )

Population: $\sigma$ Sample: s

${\sigma }^2=\frac{1}{N}\sum _{i=1}^N(x_i-\mu {)}^2$ $s^2=\frac{1}{n-1}\sum _{i=1}^n(x_i-\bar{x}{)}^2$ Once rearranged: Incremental and efficient computation of variance:

Note:

Rearranging the formula to ${\sigma }^2=\frac{1}{N}\cdot (\sum x_i^2-N{\mu }^2)$ enables you to add datapoints incrementally

• This allows for incremental and efficient computation of variance, as the sum of squares (Σx_i^2) can be updated with each new data point.

![INFO] Note This video explains why we divide by $n-1$ in sample populations

Graphical Displays

Boxplot - 5 number summary (min, $Q_1$, median, $Q_3$, max) Histogram - x-axis are values, y-axis represent frequencies Quantile plot - each value $x_i$ is paired with $f_i$ indicating that approximately 100$f_i\%$ of data are $\le x_i$ Quantile-quantile plot: graphs quantiles of univariant distribution against quantiles of another Scatter Plot: pair of values plotted as points on a plane

Boxplot

$Q_1$ = 25th percentile $Q_2$ = 75th percentile

IQR $=Q_1-Q_3$

Outlier - usually, a value higher/lower than $1.5\times$ IQR $from{Q}_3/Q_1$

Bar Chart

Plots categorical quantitative data

Histogram

Shows distributions of variables represented by area Plot binned quantitative data

Quantile Plot

Plots quantile information for all data

Quantile-quantile plot

Graphs quantiles of one univariate distribution against the corresponding quantiles of another

If the data distribution is close to normal, the plotted points will lie close to a sloped straight line

Scatter Plot